Counterexample to a variant of the Hanani-Tutte theorem on the surface of genus 4

Jan Kyncl

KAM

Abstract

We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be generalized to the orientable surface of genus 4. As a base step in the construction we use a counterexample to the unified Hanani-Tutte theorem on the torus.

The result was obtained together with Radoslav Fulek.